International Association for Cryptologic Research

International Association
for Cryptologic Research


Paper: On the Security of HFE, HFEv- and Quartz

Nicolas T. Courtois
Magnus Daum
Patrick Felke
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Abstract: Quartz is a signature scheme based on an HFEv- trapdoor function published at Eurocrypt 1996. In this paper we study "inversion" attacks for Quartz, i.e. attacks that solve the system of multivariate equations used in Quartz. We do not cover some special attacks that forge signatures without inversion. We are interested in methods to invert the HFEv- trapdoor function or at least to distinguish it from a random system of the same size. There are 4 types of attacks known on HFE: Shamir-Kipnis, Shamir-Kipnis-Courtois, Courtois, and attacks related to Gr\"{o}bner bases such as the F5/2 attack by Jean Charles Faug\`{e}re. No attack has been published so far on HFEv- and it was believed to be more secure than HFE. In this paper we show that even modified HFE systems can be successfully attacked. It seems that the complexity of the attack increases by at least a factor of $q^{tot}$ with $tot$ being the total number of perturbations in HFE. From this and all the other known attacks we will estimate what is the complexity of the best "inversion" attack for Quartz.
  title={On the Security of HFE, HFEv- and Quartz},
  booktitle={IACR Eprint archive},
  keywords={public-key cryptography / asymmetric cryptography, finite fields, multivariate cryptanalysis, Gr\"{o}bner bases, Hidden Field Equation, HFE problem, Quartz, Nessie project},
  note={- 11947 received 12 Sep 2002, last revised 17 Sep 2002},
  author={Nicolas T. Courtois and Magnus Daum and Patrick Felke},